This week I dived into the math blogosphere and found this cool blog Matt Baker’s Math Blog by Dr. Matt Baker, a professor, and Associate Dean at Georgia Tech School of Mathematics. This blog was featured back in 2013 in Evelyn’s post “How Quadratic Reciprocity Is Like Dealing Cards“. There she talks about a blog post in which Baker uses a deck of cards to describe quadratic reciprocity, a theorem in modular arithmetic that gives condition for when it’s possible to solve a quadratic equation modulo prime numbers. What caught my attention about this blog is that it has been active since 2013 and covers a wide breadth of topics including but not limited to “number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics”. As described in the about me page by Baker himself,
“Many of my recent papers are kind of long, and I’m hoping to post overviews of what’s in them and why a person might hypothetically care. I also want to post some new perspectives on older papers of mine, for example streamlined proofs or links to related work. The blog won’t be just about my own work, though: I also want to highlight recent preprints that I find exciting and share my thoughts on them. In addition, I hope to revive some old chestnuts from the past which I think deserve to be better known. I also want to share some thoughts about teaching in the 21st century with the hope of starting interesting and/or valuable dialogues. Finally, I hope to share some of the simple joys I find in math problems with beautiful solutions or things that are just plain fun. So hopefully there will be something for everyone in this blog — well, not everyone but you know what I mean.”
In this post, I will give a glimpse of some of his most recent posts.
Mental Math and Calendar Calculations
In this post, Baker talks about the many different systems to mentally calculate the day of the week on any given date. He reflects on a discussion he had with John Conway, about the pros and cons of these systems. Here he covers two systems, the Gauss-Zeller algorithm (i.e. Day of the Week = Month code + Day + Year Code + Century Adjustment (modulo 7)). and Conway’s Doomsday Method (i.e. Day of the week for Doomsday = Year Code + Century Adjustment (modulo 7)). Both these methods rely on encoding the year and century adjustment of the date. As he mentions, calculating the year code is one of the most intensive aspects of these methods and provides alternatives to speed up the calculations such as the Lewis Carroll’s method, Mike Walters’s “Easy Doomsday” method, The “Odd + 11” method which he describes in detail. What I enjoyed the most about this post were the many examples to practice mental calculations and the detailed explanations of each method.
Colorings and embeddings of Graphs
As a tribute to colleague and friend Robin Thomas who passed away last March from Amyotrophic lateral sclerosis (ALS). In this post, Baker shares some personal remarks about his friendship with Thomas and two of his most famous theorems. This post gives a glimpse of Robin Thomas beyond his math which I deeply appreciated and it’s a great way to remember him by. The first theorem he tackles is Thomas, Robertson and Seymour’s 1993 proof of the Hadwiger’s conjecture, which is a generalization of the four-color theorem, for graphs without a $K_6$-minor. The second theorem which he highlights is their classification of the forbidden minors for linklessly embeddable graphs. He states, “a graph is called intrinsically linked if every embedding in $\Bbb R^3$ contains a pair of linked cycles, and linklessly embeddable otherwise.” In this description of the theorem, he explains that the Petersen family of graphs that are intrinsically linked provides a link (pun intended) between a connection the minors of a graph and a graph being linklessly embeddable. Mainly, he states that “the theorem of Robertson-Seymour-Thomas asserts, conversely, that a graph with no minor belonging to the Petersen family is linklessly embeddable.”
As a parting thought, he remarks,
“Of course, I’ve barely scratched the surface here, both in terms of Robin Thomas’s contributions to mathematics (he published over 115 papers from 1984 to 2019) and on the subjects of graph colorings and graph embeddings. But I hope this little panoply helps highlight some of the marvelous contributions of Robin Thomas (and John Conway) to the subject.”
The Balanced Centrifuge Problem
I enjoyed reading this 2018 blog post with a neat biological application. Baker recounts chatting with a cancer researcher, Iswar Hariharan, and learning about an interesting problem he had been thinking about for a while. Centrifuges, a laboratory device that separates liquids by density by spinning test tubes, must be balanced to avoid being damaged. In this context, balanced means that “the center of mass of the collection of test tubes coincides with the center of mass of the centrifuge itself”. He poses the following question,
“If you spend a lot of time balancing centrifuges and have a mathematically curious mind, the following question might naturally arise: For which pairs (n,k) with 1 ≤ k ≤ n can you find a way to balance k identical test tubes in an n-hole centrifuge?”
Throughout the post, he provides the details on some special cases of configurations of test tubes, discusses Iswar’s conjecture which states that “you can balance k identical test tubes, 1 ≤ k ≤ n, in an n-hole centrifuge if and only if both k and n-k can be expressed as a sum of prime divisors of n”. Curiously, by translating the question into a problem about linear relations between roots of unity he found it was proven in 2010 by Gary Sivek.
He is also a mathemagician and author of ‘The Buena Vista Shuffle Club’, a book dedicated to magic tricks. I took a look at the introduction and found a great description of his magic.
“My magic tends to appeal more to the mind than to the eyes. It’s primarily card magic, frequently with some kind of mathematical principle happening in the background. But I try not to limit myself by viewing these general characteristics as constraints; on the contrary, I’m constantly testing boundaries4 to see if I can challenge myself with something unfamiliar. If you’re willing to come along for the journey, I hope you’ll enjoy the diversity of effects and methods which you’ll find in these pages.”
As he mentions in the article ‘The Magic of Math’: “There’s a lot of math in card magic,” he said. “Just like with a recipe, you might be able to follow the recipe and execute it, but you may not know enough about how it works to vary it. With card magic, I know enough to be able to combine principles in new ways and jazz around with existing effects.” Many times mathematics has seemed truly magical to me. Through his blog or his magic, Baker takes us through a pretty neat journey of mathematical discovery.
Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Resources to share? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)